Asymptotic behavior of the stochastic heat equation over large intervals

2025/09/02 by David Candil, Candil, David, Robert C. Dalang +3
Economics, Econometrics and Finance · Mathematics · #35A35 #35K05 #35K08 #35R60 #FOS: Mathematics #Nonlinear Differential Equations Analysis #Primary 60H15 #Probability (math.PR) #Secondary 60H20 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2509.02504

openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a nonlinear stochastic heat equation on [0,T]× [-L,L], driven by a space-time white noise W, with a given initial condition u₀: ℝ → ℝ and three different types of (vanishing) boundary conditions: Dirichlet, Mixed and Neumann. We prove that as L→∞, the random field solution at any space-time position converges in the L^p(Ω)-norm (p≥ 1) to the solution of the stochastic heat equation on ℝ (with the same initial condition u₀), and we determine the (near optimal) rate of convergence. The proof relies on estimates of differences between the corresponding Green's functions on [-L, L] and the heat kernel on ℝ, and on a space-time version of a Gronwall-type lemma.

Citations

• Stochastic Partial Differential Equations, Space-time White Noise and Random Fields
• Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds

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